Do I pay Zakah on employee stock or stock options? Does it matter if they are vested, qualified, or restricted?

Short answer

You do not have a right to unvested employee stock so there is no Zakah on that. Vested employee stock options cannot be sold so there is no Zakah on them until you exercise the option and get the actual shares. Vested (restricted or unrestricted) stock from an employer is similar to shares that you purchase in the market, so Zakah is due on them.